\newproblem{lay:2_1_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 2.1.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Suppose the third column of $B$ is the sum of the first two columns. What can be said about the third column of the product $AB$?
}{
  % Solution
	Let us consider the different columns of $B$
	\begin{center}
	   $B=\begin{pmatrix}\mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 & ... \end{pmatrix}$
	\end{center}
	The product of $AB$ is
	\begin{center}
		$AB=A\begin{pmatrix}\mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 & ... \end{pmatrix}=\begin{pmatrix}A\mathbf{b}_1 & A\mathbf{b}_2 & A\mathbf{b}_3 & ... \end{pmatrix}$
	\end{center}
	If $\mathbf{b}_3=\mathbf{b}_1+\mathbf{b}_2$, then
	\begin{center}
		$A\mathbf{b}_3=A(\mathbf{b}_1+\mathbf{b}_2)=A\mathbf{b}_1+A\mathbf{b}_2$
	\end{center}
	That is, the third column of $AB$ is also the sum of the first and second columns of $AB$.
}
\useproblem{lay:2_1_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
